Signal Detection Theory and Psychophysics - Andrei Gorea

d’ with unequal varianceszH210-1-2d’ a'd 1in s N units'd Y / Nd-2 -1 0 1 2zFA''a 2dY / N R/d’ a'd2in s S unitss21 sThe index d a has the properties we want.221. It is intermediate in size between d 1 and d‘ 2 .2. It is equivalent to d' when the ROC slope is 1,for then the perpendicular line of length D Y/Ncoincides with the minor diagonal, and the twolines of length d a coincide with the d' 1 , and d' 2segments.3. It turns out to be equivalent to the differencebetween the means in units of the root-meansquare(rms) standard deviation, a kind ofaverage equal to the square root of the mean ofthe squares of the standard deviations of S1and S2.To find d a from an ROC that is linear in z coordinates, it iseasiest to first estimate d’ 1 , d’ 2 and the slope s = d’ 1 /d’ 2 .Because the standard deviation of the S1 distribution is stimes as large as that of S2, we can set the standarddeviation of S1 to s and that of S2 to 1.To find d a directly from one point on the ROC once s is known:dd'a'ad'2.5 1s²zH sz.5F d'2 2 1s² 2 1s² .5.5